Towards a comprehensive physically-based rainfall-runoff model

Towards a comprehensive physically-based rainfall-runoff model

859

Hydrology and Earth System Sciences, 6(5), 859–881 (2002) © EGS

Towards a comprehensive physically-based rainfall-runoff modelZhiyu Liu and Ezio Todini

Department of Earth and Geo-Environmental Sciences, University of Bologna, Via Zamboni 67, 40127 Bologna, Italy

Email for corresponding author: [email protected]

AbstractThis paper introduces TOPKAPI (TOPographic Kinematic APproximation and Integration), a new physically-based distributed rainfall-runoff model deriving from the integration in space of the kinematic wave model. The TOPKAPI approach transforms the rainfall-runoff andrunoff routing processes into three ‘structurally-similar’ non-linear reservoir differential equations describing different hydrological andhydraulic processes. The geometry of the catchment is described by a lattice of cells over which the equations are integrated to lead to acascade of non-linear reservoirs. The parameter values of the TOPKAPI model are shown to be scale independent and obtainable from digitalelevation maps, soil maps and vegetation or land use maps in terms of slope, soil permeability, roughness and topology. It can be shown,under simplifying assumptions, that the non-linear reservoirs aggregate into three reservoir cascades at the basin scale representing the soil,the surface and the drainage network, following the topographic and geomorphologic elements of the catchment, with parameter valueswhich can be estimated directly from the small scale ones. The main advantage of this approach lies in its capability of being applied atincreasing spatial scales without losing model and parameter physical interpretation. The model is foreseen to be suitable for land-use andclimate change impact assessment; for extreme flood analysis, given the possibility of its extension to ungauged catchments; and last but notleast as a promising tool for use with General Circulation Models (GCMs). To demonstrate the quality of the comprehensive distributed/lumped TOPKAPI approach, this paper presents a case study application to the Upper Reno river basin with an area of 1051 km2 based on aDEM grid scale of 200 m. In addition, a real-world case of applying the TOPKAPI model to the Arno river basin, with an area of 8135 km2

and using a DEM grid scale of 1000 m, for the development of the real-time flood forecasting system of the Arno river will be described. TheTOPKAPI model results demonstrate good agreement between observed and simulated responses in the two catchments, which encouragesfurther developments of the model.

Keywords: rainfall-runoff modelling, topographic, kinematic wave approximation, spatial integration, physical meaning, non-linear reservoirmodel, distributed and lumped

IntroductionThe study of the impacts of land use and climate changeson the hydrological regimes of river basins requires a betterunderstanding of how climate, topography-geomorphology,soils and vegetation interact to control runoff at the field,hillslope and catchment scales. These interactions can berepresented within complex physically-based distributedmodels (e.g. SHE: Abbott et al., 1986 a,b). But, traditionalphysically-based distributed models usually work at a smallsize and require a large amount of data and lengthycomputation times which limit their application. Todini(1988) has pointed out that a promising direction in modeldevelopment is to lump the differential equations atincreasing scales except for a few essential parameters andto make them computationally affordable. Although thevalidity of effective parameter values that must be used with

large scale catchments has been questioned (Beven, 1989),nonetheless a correct integration of the differential equationsfrom the point to the finite dimension of a pixel, and fromthe pixel to larger scales, can actually generate relativelyscale-independent physically- based models, which preservethe physical meaning (although as averages) of the modelparameters.

On the basis of a critical analysis of two well known andwidely used hydrological rainfall-runoff models - namelythe ARNO (Todini, 1996) and the TOPMODEL (Beven andKirby, 1979; Beven et al., 1984; Sivapalan et al., 1987),Todini (1995) recently proposed the TOPKAPI(TOPographic Kinematic APproximation and Integration)model. The ARNO model is a variable contributing areasemi-distributed conceptual model controlled by the totalsoil moisture storage, and widely used for real-time flood

Zhiyu Liu and Ezio Todini

860

forecasting. The major disadvantage of the ARNO modelis the lack of physical grounds for establishing some of theparameters, which reduces its possible extension toungauged catchments. The TOPMODEL is a variablecontributing area model in which the predominant factorsdetermining the formation of runoff are represented by thetopography, the transmissivity of the soil and its verticaldelay. However, the model preserves its physical meaningonly at the hillslope scale (Franchini et al., 1996), while itdegrades into a conceptual model at larger scales, with thesame problems mentioned for the ARNO model.

In contrast, the TOPKAPI model is based on the lumpingof a kinematic wave assumption in the soil, on the surfaceand in the drainage network, and leads to transforming therainfall-runoff and runoff routing processes into three non-linear reservoir differential equations which can be solvedanalytically (Liu, 2002). The geometry of the catchment isdescribed by a lattice of cells (the pixels of a DEM) overwhich the equations are integrated to lead to a cascade ofnon-linear reservoirs. The parameter values of the TOPKAPImodel are scale independent and obtainable from a digitalelevation map, soil map and vegetation or land-use map interms of slope, soil permeability, roughness and topology.It can be shown, under simplifying assumptions, that thenon-linear cascade aggregates into a unique non-linearreservoir at the basin level, with parameter values that canbe estimated directly from the small scale ones withoutlosing physical meaning.

The present paper explains the overall structure andmethodology of the TOPKAPI model, the non-linearreservoir equation and its solution technique, the datarequirements, and the model calibration procedure. Todemonstrate the quality of the TOPKAPI approach, a casestudy of applying the TOPKAPI model to the Upper Renoriver basin is presented, to clarify the data and calibrationrequirements of the model together with the aggregationcapabilities when moving from the distributed to the lumpedversions of the TOPKAPI model. A real-world case ofapplying the TOPKAPI model to the Arno river basin, withan area of 8135 km2 and using a DEM grid scale of 1000 m,for the development of areal-time flood forecasting systemfor the Arno River is also presented.

Structure and methodology of theTOPKAPI modelThe TOPKAPI is a comprehensive distributed-lumpedapproach. The distributed TOPKAPI model is used toidentify the mechanism governing the dynamics of thesaturated area contributing to the surface runoff as a functionof the total water storage, thus obtaining a law underpinning

the development of the lumped model.The model is based on the idea of combining the kinematic

approach with the topography of the basin; the latter isdescribed by a Digital Elevation Model (DEM) whose gridsize generally increases with the overall dimensions. Eachgrid cell of the DEM is assigned a value for each of thephysical characteristics represented in the model. The flowpaths and slopes are evaluated starting from the DEM,according to a neighbourhood relationship based on theprinciple of minimum energy cost (Band, 1986).

The integration in space of the kinematic wave equationsresults in three ‘structurally-similar’ non-linear reservoirequations describing different hydrological and hydraulicprocesses. This lumping is performed on the individual cellof the DEM in the distributed model, while in the lumpedmodel it is performed at the basin level. The equationsobtained for the local scale and for the lumped scale arestructurally similar; what distinguishes them are thecoefficients, which in one case have local significance and,in the other, summarise the local properties in a globalmanner.

The present TOPKAPI model (Fig. 1) is structured aroundfive modules that represent the evapotranspiration,snowmelt, soil water, surface water and channel watercomponents respectively. For the deep aquifer flow, theresponse time caused by the vertical transport of waterthrough the thick soil above this aquifer is so large thathorizontal flow in the aquifer can be assumed to be almostconstant with no significant response on one specific stormevent in a catchment (Todini, 1995). Hence, initially, themodel does not account for water percolation towards thedeeper subsoil layers and for their contribution to thedischarge; this is planned as an additional model layer in afuture model development.

The soil water component is affected by subsurface flow(or interflow) in a horizontal direction defined as drainage;drainage occurs in a surface soil layer, of limited thicknessand with high hydraulic conductivity due to its macro-porosity. The drainage mechanism plays a fundamental rolein the model both as a direct contribution to the flow in thedrainage network and most of all as a factor regulating thesoil water balance, particularly in activating the productionof overland flow. The soil water component is the mostcharacterising aspect of the model because it regulates thefunctioning of the contributing saturated areas. The surfacewater component is activated on the basis of this mechanism.Lastly, both components contribute to feeding the drainagenetwork.

The most complex and physically realistic model forestimating actual evapotranspiration is the Penman-Monteithequation, which has been widely used in many distributed


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Fig. 1. The components of the TOPKAPI model

models, e.g. SHE (Abbott et al., 1986 a,b), DHSVM(Wigmosta et al., 1994). However, a simplified approachis generally necessary because in many countries therequired historical data for Penman-Monteith estimationsare not extensively available; and, in addition, apart from afew meteorological stations, almost nowhere are real-timedata available for flood forecasting applications. Evapo-transpiration plays a major role not only in terms of itsinstantaneous impact, but in terms of its cumulative temporaleffect on the soil moisture volume depletion; this reducesthe need for an extremely accurate expression, provided thatits integral effect is well preserved. In the present TOPKAPImodel, evapotranspiration can be introduced directly as aninput to the model or computed externally or estimatedinternally by a radiation method (Doorembos et al., 1984)starting from the temperature and from other topographic,geographic and climatic information, as described in theARNO model (Todini, 1996). Again, for reasons of limiteddata availability, the snow accumulation and melting(snowmelt) component is driven by a radiation estimatebased upon the air temperature measurements, which is alsoborrowed from the ARNO model. In the following sections,the three basic components, the soil water, surface waterand channel water will be described in detail.

The distributed TOPKAPI modelTHE SOIL WATER COMPONENT

Fundamental assumptions

(1) Precipitation is constant over the integration domain(namely the single cell), by means of suitable averagingand lumping operations of the local rainfall data, suchas Thiessen polygons and Block Kriging (de Marsily,1986; Matheron, 1970) techniques;

(2) All of the precipitation falling on the soil infiltrates intoit, unless the soil is already saturated in a particular zone;this is equivalent to adopting as the sole mechanism forthe formation of overland flow the saturationmechanism from below, (Dunne, 1978);

(3) The slope of the water table coincides with the slope ofthe ground, unless the latter is very small (less than0.01%); this constitutes the fundamental assumption ofthe approximation of the kinematic wave in the SaintVenant equations, and it implies the adoption of akinematic wave propagation model with regard tohorizontal flow, or drainage, in the unsaturated area(Henderson and Wooding, 1965; Beven, 1981);

(4) Local transmissivity, like local horizontal flow, dependson the total water content of the soil, i.e. it depends on

Meteorological data(precipitation, air temperature)

DEM Soil type Landuse

Evapotranspiration Model

Snowmelt Model

Soil Water Model

Surface Water ModelPercolation and Groundwater Model

Channel Water Model Lake and Reservoir Model


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the integral of the water content profile in a verticaldirection;

(5) Saturated hydraulic conductivity is constant with depthin a surface soil layer but is much larger than that atdeeper layers; this forms the basis for the verticalaggregation of the transmissivity.

Kinematic wave formulation for sub-surface flow

In the TOPKAPI model, the horizontal sub-surface flow, q,is calculated by the approximation Eqn. (1) (Benning, 1994;Todini and Ciarapica, 2001):

( ) αβ Θ= ~tan Lkq s(1)

with ( ) ∫∫ −−

==ΘL

rs

rL

dzL

dzzL 00

1~1~ϑϑϑϑ

ϑ

where β is the slope angle, ks is the saturated hydraulicconductivity in ms-1, L is the thickness of the surface soillayer in m, Θ~ is the reduced soil moisture content, rϑ isthe residual soil moisture content, sϑ is the saturated soilmoisture content, ϑ is the water content in the soil, ( )zϑ~ isthe mean value along the vertical profile of the reduced soilmoisture content and α is a parameter which depends onthe soil characteristics (Benning, 1994; Todini, 1995).

Combined with the equation for continuity of mass, thefollowing system is obtained:

( )

( )

Θ=

=+Θ−

αβ∂∂

∂∂ϑϑ

~tan

~

Lkq

pxq

tL

s

rs (2)

where x is the main direction of flow along a cell, t is thetime, q is the horizontal flow in the soil due to drainage,corresponding to a discharge per unit of width in m2 s–1, andp is the intensity of precipitation in m s–1. The model iswritten in just one direction since it is assumed that the flowis characterised by a preferential direction, which can bedescribed as the direction of maximum slope.

Equation (2), rewritten in terms of actual total watercontent in the soil, ( ) Θ−= ~Lrs ϑϑη , along the verticalprofile, with the substitution

( )( ) ααϑϑ

βL

LkCrs

s−

= tan (3)

leads to the kinematic equation

( )( ) x

CpxL

kLpt rs

s

∂η∂

∂η∂

ϑϑβ

∂η∂ αα

αα −=−

−= tan. (4)

Here, the term C represents in physical terms a localconductivity coefficient, since it depends on soil parametersfor a particular position or a particular cell, whichencompasses the effects of hydraulic conductivity and slope,to which it is directly proportional, and storage capacity, towhich it is inversely proportional.

Non-linear reservoir model for the soil water in a genericcell

By integrating Eqn. (4) in the soil over the ith DEM gridcell, whose space dimension is X, gives

( )s

ii

s

ii

iss

s CCpXtv αα ηη∂∂

11 −−−−= (5)

where isv is the volume per unit of width stored in the ith

cell in m2, while the last term in Eqn. (5) represents theinflow and outflow balance. A subscript s is introduced hereto distinguish this soil water equation from the ones relevantto the overland and the drainage network flows.

It is quite evident that the coefficients CS are no longerthe physically measurable quantities, which are defined ata point; rather, they represent integral average values forthe entire cell, which nonetheless are still strongly relatedto the measurable quantities.

In the TOPKAPI model, the grid cells are connected by atree shaped network; water moves downslope along this tree-shaped flow pathway starting from the initial cells (withoutupstream contributing areas) representing the ‘sources’,towards the outlet. According to this procedure, andassuming that in each cell the variation of the vertical watercontent along the cell is negligible, the volume of waterstored in each cell (per unit width) can be related to thetotal water content that is equivalent to the free water volumein depth, by means of the simple expression

iiXvs η= (6)

Substituting for ηi in Eqn. (6) and writing it for the“source” cells in each flow pathway, the following non-linear reservoir equation is obtained:

s

is

iis

si

s VX

XCXp

tV α

α∂∂

22 −= . (7)

Similarly a non-linear reservoir equation can be writtenfor a generic cell, given the total inflow to the cell, suchthat

( ) s

is

i

ii

is

sus

uoi

s VX

XCQQXp

tV α

α∂∂

22 −++= (8)

( ) pxq

tLrs =+Θ−

∂∂

∂∂ϑϑ

~


863

where isV is the volume stored in the ith cell in m3, u

oiQ is

the discharge entering the active cell i as overland flow fromthe upstream contributing area in m3 s–1, and u

siQ is the

discharge entering the active cell as subsurface flow fromthe upstream contributing area in m3 s–1.

Soil moisture accounting in a grid cell

For the ith cell at each time-step, the soil water balance canbe calculated as follows:

T

tVTtVQQXpQ ii

iii

ssus

uoi

ds

)()()( 00

'2 −+

−++= (9)

( )[ ]{ }0,),('min)('max 00 iiii ssso VTtVTtVe +−+= (10)

[ ] 200 ),('min)( XEVTtVTtV asmss iii

−+=+ (11)

where T is the computational time interval in seconds, t0 isthe initial time of the computation step, )( 0

' TtVis + is the

solution of Eqn. (8) at the time t0 + T in m3, )( 0tVis is the

initial soil water storage in the ith cell at the time t0 in m3 ,dsi

Q is the outflow discharge from the ith cell during the timet0 to t0 + T in m3 s–1,

ioe is the saturation excess volume forthe ith cell in m3,

ismV is the saturated soil water storage inthe ith cell in m3, Ea is the actual evapotranspiration withinthe time interval calculated by the evapotranspiration modelin m, and )( 0 TtV

is + is the soil water volume stored in thesoil in the ith cell at time t0 + T.

Up to this point it has been implicitly assumed that theentire outflow from a cell flows into the downstream cellimmediately. However, this is not entirely true since notehas to be taken of the depletion effected by the drainagenetwork. Thus, for the cells in the channel network, theoutflow is still evaluated by Eqn. (9), but it is then partitionedbetween the channel and the downstream cell according toa gradient based upon the average slope of the foursurrounding cells. This allows determination of the amountof subsurface flow feeding the drainage channel network.This operation of flow partition is also performed for theoverland flow.

THE SURFACE WATER AND CHANNEL WATERCOMPONENTS

The input to the surface water model is the precipitationexcess resulting from the saturation of the surface soil layer.In addition, the water in the soil can ex-filtrate on the surfaceas return flow due to a sudden change in hillslope or soil

property, and thus also feeds the overland flow. The sub-surface flow and the overland flow together feed the channelalong the drainage network.

Non-linear reservoir model for overland flow and channelflow in a grid cell

Overland flow routing is described similarly to the soilcomponent, according to the kinematic approach (Wooding,1965) in which the momentum equation is approximatedby means of Manning’s formula. The kinematic waveapproximation for overland flow is described as

==

−=

oooo

oo

oo

o

hChn

q

xqr

th

αβ

∂∂

∂∂

35

21

)(tan1(12)

where oh is the water depth over the ground surface in m,or is the saturation excess resulting from the solution of the

soil water balance, either as the precipitation excess or theex-filtration from the soil in absence of rainfall in ms-1, onis the Manning friction coefficient for the surface roughnessin 13

1 −− sm , oo nC 21

)(tan β= is the coefficient relevant tothe Manning formula for overland flow, and 35=oα is theexponent which derives from using the Manning formula.A subscript o denotes the overland flow.

By analogy with what was done for the soil, assumingthat the surface water depth is constant over the cell andintegrating the kinematic equation over the longitudinaldimension, gives the non-linear reservoir model for theoverland flow for the ith cell as

o

io

i

i

io

oo

o VX

XCXr

tV α

α∂∂

22 −=

(13)where

ioV is the surface water volume in the cell in m3.Similar considerations also apply to the channel network,

which is assumed to be tree-shaped with reaches havingwide rectangular cross-sections. In this case, the channelsurface width is not constant but is assumed to be increasingtowards the catchment outlet. Under these assumptions, thenon-linear reservoir model can be written for a generic reachas

( )( )

c

icii

ic

i

iicucic

c VXW

WCQXWr

tV α

α−+=∂∂ (14)

where icV is the volume of water stored in the ith channel

reach in m3, iW is the width of the ith rectangular channelreach in m, u

ciQ is the inflow discharge from the upstream


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reaches in m3 s–1, cr is the lateral drainage input, includingthe overland runoff reaching the channel reach and the soildrainage reaching the channel reach in m s–1, cc nsC 2

10=

is the coefficient relevant to the Manning formula forchannel flow, s0 is the bed slope, assumed to be equal to theground surface slope, cn is the Manning friction coefficientfor the channel roughness in 13

1 −− sm and 35=cα is theexponent which derives from using the Manning formula.The channel width Wi is taken to increase as a function ofthe area drained by the i th cell on the basis ofgeomorphological considerations, such that

( )totdrthtot

minmaxmaxi AA

AAWW

WWi−

−−

+= (15)

where maxW is the maximum width, at the basin outlet, minWis the minimum width, corresponding to the threshold area,Ath is the threshold area which is the minimum upstreamdrainage area required to initiate a channel, Atot is the totalarea and

idrA is the area drained by the ith cell.

PHILOSOPHY OF APPLICATION OF THEDISTRIBUTED TOPKAPI MODEL

Solution technique for the non-linear reservoir equation

The TOPKAPI formulation leads to three non-linearreservoir equations describing the subsurface flow, theoverland flow and the channel flow. In the first version ofthe TOPKAPI model (Todini and Ciarapica, 2001), thesolution of the non-linear reservoir equation was based upona variable step fifth order Runge-Kutta numerical algorithmdue to Cash and Karp (1990). Nowadays, the authors havefound that the non-linear reservoir equation can be solvedanalytically based on an appropriate approximation (Liu,2002), providing a more efficient scheme in terms of runningtime. Details of analytical solutions based on appropriateapproximations are given in Appendix A.

Data requirements and parameters

The TOPKAPI model requires terrain data (e.g. from DTMor DEM land survey), soil survey and vegetation or land-use, as well as geographical co-ordinates and measurementsof precipitation, evapotranspiration or air temperature.

As far as the parameters are concerned, there are sevenclasses of parameters in the TOPKAPI model, namely L(thickness of the surface soil layer in m), ks (saturatedhydraulic conductivity in m s–1),ϑr (residual soil moisturecontent), sϑ (saturated soil moisture content), as (exponent

of the transmissivity law for the soil component, assumedto be constant for all the cells), on (surface roughness in

131 −− sm ), and cn (roughness for the channel in 13

1 −− sm ) . Fiveof the parameters (L, k s ,ϑr , sϑ and as) relate to the soil andcontrol runoff production, whilst the other two ( on , cn ) arerouting parameters.

DEM application in the TOPKAPI model

The DEM application consists of: identifying and correctingsinks and false outlets, identifying the connections amongthe cells thereby giving the flow pathways, calculating thesteepness, and cumulating the drained area for the automaticdetection of the drainage network. Utilities serving thesepurposes are provided in GRASS (Geographic ResourcesAnalysis Support System), but unfortunately they are notsuitable for application to the TOPKAPI model because theyare based on neighbourhood functions that use a moving3 × 3 point spatial window, where the elevation of the centralpoint is compared with the heights of the eight connectedneighbours. For the TOPKAPI model, or in general for anymodel using a finite difference approach, in the moving 3 × 3point spatial window the four neighbouring cells adjacentto each corner have to be neglected, which means physicallythat drainage is only possible to the north, east, south orwest for the four adjacent cells at each edge.

Model calibration

The TOPKAPI model is considered as a physically basedmodel, with all parameters having physical meanings whichcan be measured directly through fieldwork. Although it isphysically based, the model still needs calibration becauseof the uncertainty of the information on the topography, soilcharacteristics and land cover. Nonetheless the calibrationof the TOPKAPI parameters is more an adjustment than aconventional calibration and is carried out by a simple trial-and-error method.

In the TOPKAPI model, the initial soil saturation conditioncan be set as follows. First, the initial soil saturationpercentage is set as the same value (e.g. 0.5) for all cellsaccording to a dry situation at the beginning of thecalibration period. Then the TOPKAPI program is run toobtain the soil saturation condition at a chosen time whenthe soil is supposed to be in the same state as the initialcondition. This procedure gives realistic values for the initialcondition, particularly with regard to the spatial distributionof the saturation percentage. At the initial condition, it isalso assumed that there is no snow or surface water over theslopes and the water depth in a generic channel cell increasesas a linear function of the channel width of the cell.


865

Model parameter values are assigned according to the typeof soil, land cover and channel order by the method ofStrahler (1957). The initial parameter values can be takenfrom the literature, e.g. the values of the soil parameters ks ,ϑr and sϑ can be taken from the USDA parameters for theinfiltration model of Green-Ampt. Initial values forparameters no and nc can be estimated by referring to Tables5.5 and 5.6 in the book on Open Channel Hydraulics byChow (1959) and the report on Roughness Characteristicsof Natural Channels by Barnes (1967), while as usuallyvaries 2.0 ~ 4.0 based on the soil property. In general, theranges of the values of the model parameter are: L=0.10~2.00 m, k s = 10-6 ~ 10–3 m s–1, ϑs = 0.25~0.70, ϑr =0.01~0.10, on = 0.05~0.40 m–1/3 s–1, cn =0.02~0.08 m–1/3 s–1.

The lumped TOPKAPI modelDESCRIPTION OF THE LUMPED TOPKAPI MODEL

The overall structure of the lumped TOPKAPI model isshown schematically in Fig. 2. A catchment is regarded asa dynamic system composed of three reservoirs: the soil

reservoir, the surface reservoir and the channel reservoir.In the flow simulation, on the basis of the soil conditionand actual evapotranspiration, the precipitation in thecatchment is partitioned into direct runoff and infiltrationusing the Beta-distribution curve, which reflects the non-linear relationship between the soil water storage and thesaturated contributing area in the basin. The infiltrationand direct runoff are input into the soil reservoir and surfacereservoir, respectively. Outflows from the two reservoirsas interflow and overland flow are then drained into thechannel reservoir to form the channel flow.

As mentioned before, the TOPKAPI approach is acomprehensive distributed-lumped approach. Theoreticallyit proves that the lumped version of the TOPKAPI modelcan be derived directly from the results of the distributedversion and does not require additional calibration.

In order to obtain the lumped version of the TOPKAPI,the point kinematic wave equation is integrated over theentire system of cells describing the basin. This is done firstby computing the total volume stored in the soil, on thesurface or in the channel network by adding up the singlecell volumes as a function of the geomorphology and

o

oTVbARR

tV

ordoT α−+=∂∂ )(

SsTs

sT VbRAt

V α−=∂∂

ccTcso

cT VbQQt

V α−+=∂∂

)(

Runoff generation by

saturation-excesss

Return flow calculation model

SSuurrffaaccee rreesseerrvvooiirr

SSooiill rreesseerrvvooiirr

CChhaannnneell rreesseerrvvooiirr

Intteerrffllooww

Qs

OOvveerrllaanndd ffllooww Qo

EExxffiillttrraattiioonn

Rate Rr DDiirreecctt rruunnooffff Intensity Rd

Infiltration rate R

CChhaannnneell ffllooww

Net

rainfall

(P-Ea)

Fig. 2. Structure of the lumped TOPKAPI model


866

topology of the catchment. Under the assumption that thedifference between the inflow to a cell and the time variationof the water storage do not vary significantly in space, thisintegration and aggregation results in a non-linear reservoirequation for representing the basin as a whole of the form(Todini and Ciarapica, 2001; Liu, 2002).

(16)

with

where i is the index of a generic cell, j is the index of cellsdrained by the ith cell, N is the total number of cells in theupstream contributing area, sTV is the water storage in thecatchment in m3, R is the infiltration rate in m s–1, A is thecatchment area,

isC is the local conveyance for the ith cell,fm represents the fraction of the total outflow from the mth

cell which flows towards the downstream cell, αs is a soilmodel parameter assumed constant in the catchment, and

sb is a lumped soil reservoir parameter which incorporatesin an aggregated way the topography and physical propertiesof the soil.

Equation (16) corresponds to a non-linear reservoir modeland represents the lumped dynamics of the water stored inthe soil. The same type of equation can be written for theoverland flow and for the drainage network, thustransforming the distributed TOPKAPI model into a lumpedmodel characterised by three ‘structurally similar’ non-linearreservoirs, namely ‘soil reservoir’, ‘surface reservoir’ and‘channel reservoir’.

The infiltration rate term in Eqn. (16) must be evaluatedby separating precipitation into direct runoff and infiltrationinto the soil. In order to obtain this separation, a relationshipbetween the extent of saturated areas and the volume stored

in the catchment is introduced, similar to what is done inthe Xinanjiang model (Zhao, 1977), in the ProbabilityDistributed Model (Moore and Clarke, 1981; Moore, 1985,1999) and in the ARNO model (Todini, 1996). Given theavailability of the distributed TOPKAPI version, thisrelationship can be obtained by means of extensivesimulation. At each time step the number of saturated cellsis compared to the total volume of water stored in the soilover the entire catchment. Denoting total water storage inthe soil by sTV , the soil water storage at the total saturationcondition by ssV , and the total saturation area by As, therelationship between the extent of saturated areas and thevolume stored in the catchment can be approximated by aBeta-distribution function curve expressed by:

(17)

where )(⋅Γ is the Gamma function, and r and s twoparameters of the Beta-distribution curve.

Surface runoff calculation

Surface runoff is the sum of direct runoff and the returnflow caused by ex-filtration. In the lumped TOPKAPImodel, the Beta-distribution function curve as expressedby Eqn. (17) is thus used to compute, in the lumped form,the overall inflow as infiltration R to the soil reservoir andthe overall saturation excess as direct runoff Rd (ms-1) enteredinto the surface reservoir by the following equation:

)( 12

12

ttAV

VV

VV

PR ss

ss

sT

ss

sTd −

−−= (18)

where 1sTV ,

2sTV represent the soil water storage in thecatchment at time t1, and t2=t1+∆t respectively where ∆t isthe computation time interval, and P represents thecatchment average net precipitation intensity (m s-1).

The soil water can ex-filtrate to generate the return flow.In the present TOPKAPI model, the return flow is estimatedbased on a limiting parabolic curve representing therelationship between the return flow discharge and thefraction of water storage in the catchment, which is estimatedbased on the points from the distributed TOPKAPI model.This parabolic curve can be expressed as:

Aa

VV

Aa

VV

Aa

AQRss

sT

ss

sTreturnr

32

2

1 ++

== (19)

ϕϕϕ dsrsr

AA sr

VV

sss

sT

11

0

)1()()()( −− −

ΓΓ+Γ= ∫

s

s

i

s

s

sN

l

N

lmm

j

l

j

lmm

Cf

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/11

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s

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iN

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1

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sssTssTsT VbRAVCX αα −=

s

N

l

N

lmm

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ssT

fXRA

tV

α

αα

∂

∑ +

∏

+−=

∂∂

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1

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11

11


867

where rR is the return flow ex-filtration rate, returnQ is the

calculated return flow discharge on the surface during thetime interval t1 ~ t2, )(5.0

21 sTsTsT VVV += is the averaged soilwater storage in m3, and a1, a2 and a3 are parametersestimated on the basis of the points from the distributedTOPKAPI model.

Accordingly, the infiltration rate into the soil within thetime interval Dt can be computed by:

AQ

ttAV

VV

VV

R returnss

ss

sT

ss

sT −−

−=

)( 12

12 (20)

The quantities R and Rd plus Rr are then input to the soilreservoir and the surface reservoir, respectively (see Fig. 3).The interflow and overland flow can be obtained bycomputing the water balance, and are then together drainedinto the channel reservoir to generate the total outflow atthe basin outlet.

Fields of applicationSince its advent in 1995, the TOPKAPI model has beenapplied to several catchments for different uses such as floodforecasting, extreme flood analysis, and predictinghydrological response under changed landscape conditionscaused by human activities. The distributed version of theTOPKAPI model allows for its calibration on the basis ofphysical considerations and, in particular, its extension toungauged catchments. The lumped version of the TOPKAPImodel allows for the extensive simulations needed whenused in combination with a stochastic rainfall generator forderiving, through continuous simulation, extreme dischargesand flood wave volumes.

The model has been applied to the upper Reno river basin(1051 km2) (Todini and Ciarapica, 2001) and the Arno riverbasin (8135 km2) for flood forecasting (Liu, 2002), and tothe Magra catchment (1682 km2) for extreme flood analysis(Liu and Todini, 2000). The DEM grid size employed

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Fig. 3. TOPKAPI model results in the Upper Reno catchment: (a) small flood events, (b) major flood events

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868

increases with the catchment size, ranging from 200 m to1 km for these three examples.

A case study applying both the distributed and lumpedTOPKAPI model to the upper Reno river basin and includingthe distributed TOPKAPI model for developing a real-timeflood forecasting system for the Arno River is presentedhere in detail.

The Upper Reno river basin casestudyCATCHMENT CHARACTERISTICS

Topography

The Reno River in Italy rises on the northern slopes of theApennines and flows inside the Emilia-Romagna region(Fig. 4a) with a total watercourse of 210 km. The basin hasa surface area of 4930 km2 and a variation in height abovesea level of up to 2,000 m (see Fig. 4b). The upper part ofthe basin is gauged at Casalecchio on the outskirts of thecity of Bologna, and drains an area of approximately1051 km2 within which it reaches its maximum elevation of2000 m. A digital elevation model (DEM) on a 400 × 400 m

grid is available for most of the Upper Reno river basinfalling under the Emilia Romagna Regional Authority.

Soils

The Upper Reno river basin comprises primarily clayey andmarly soil, as well as alluvial deposits in its lower section(see Fig. 4c and Table 1).

Table 1. Soil types in the Upper Reno river basin(Fig. 3c)

Class Soil Types

a Arenaceous turbidites, calcarenitesb Arenaceous turbidites, marly limestone turbiditesc Marly arenaceous turbiditesd Arenaceous turbiditese Marly limestone and calcarenite turbidites, clays

and marly limestonesf Clays, sands and conglomeratesg Alluvial deposits

a)

b) c) c) b)

Casalecchio

Vergato

Calcara Castenaso

Mordano

Bastia

Sesto Imolese

Fig. 4. Reno river basin: a) Reno river system, sub-basins and water level stations, b) DEM map for the Upper Reno catchment(legend: elevation above sea level in meter), c) Soil type map for the Upper Reno catchment


869

Vegetation and land use

Vegetation on the highest parts of the Apennines is madeup of herbaceous formations and blueberry derived fromcultivation and grazing which were once particularlyintensive. The most widespread land use in the strip ofterritory along the Reno, both for the lowland area as wellas for the mountain area, is agriculture, except for theimpracticable areas at high levels and those areas subject toperiodical flooding.

MODEL CALIBRATION OF THE DISTRIBUTEDTOPKAPI MODEL

Hydrometerological data availability

The hydrological dataset in the year 1990 is selected forcalibration in the Upper Reno river basin, since in this year,from 24 to 26 November, 1990, a relatively large flood eventwith a peak discharge of 1317 m3 s–1 occurred. Hourlymeasurements from 24 raingauges and 10 thermometerswere available while discharges were computed from hourlylevels by means of a well-verified rating curve availablefor Casalecchio. The areal rainfall distribution was estimatedby the Thiessen Polygon method.

Model calibration

The model calibration was performed at a 1-hour time-stepusing the hydrological dataset of 1990. An initial estimatefor the model parameter set was derived using the availablebroad descriptions of soil types given in Table 1 and Strahlerchannel order, together with values taken from the literature.Adjustment of parameters was performed manually and, atthe end, the values given in Table 2 were retained. The modelresults are shown in Fig. 3, and the model efficiencies are:0.850 (R2: the proportion of the variance in the observations

accounted for by the model), 0.843 (r2: the coefficient ofdetermination) and 0.918 (r: the correlation coefficient),respectively.

DERIVATION OF THE LUMPED TOPKAPI MODELPARAMETERS AND MODEL APPLICATION

As explained previously, the lumped TOPKAPI model canbe derived directly from the distributed results of thedistributed TOPKAPI model. Given the availability of thedistributed TOPKAPI version, at each step in time thenumber of saturated cells is compared with the total volumeof water stored in the soil over the entire catchment. Therelationship between the extent of saturated areas and thevolume stored in the catchment can be approximated by aBeta-distribution function curve which is shown in Fig. 5.

In addition, the distributed TOPKAPI model showed thatthe overland flow comprises two parts, one from directrunoff in the saturated area and the other from return flowwhich is the ex-filtration of soil water into the surface. Inthe Upper Reno river basin, the maximum discharge valueof the return flow was 43 m3 s–1. Hence, the generation ofreturn flow is very significant in the catchment, and thereforeshould be simulated in the lumped TOPKAPI model. Figure6 shows the relationship between the return flow dischargeand the soil water storage in the catchment, and itsapproximation by the limiting parabolic curve adopted inthe lumped TOPKAPI model.

The parameters of the lumped TOPKAPI model are listedin Table 3 and simulation results are shown in Figs. 7 and 8.Figure 7 compares the runoff calculated by the distributedTOPKAPI model with that obtained using the lumpedmodel, while Fig. 8 shows the comparison of the observeddischarges at Casalecchio and those computed using thedistributed and lumped TOPKAPI model. The lumpedTOPKAPI model performance measures are as high as thedistributed ones, namely R2 = 0.887, r2 = 0.867 and r = 0.931.

Table 2. Calibration parameters for the different soil classes, land uses and channel orders

Soil rs ϑϑ − α sk L (m) Soil use on Strahler cnClass (m s-1) class (m-1/3 s-1) channel (m-1/3 s-1)

order

a 0.36 2.5 2.50E-04 1.65 a 0.085 I 0.045b 0.36 2.5 2.20E-04 1.60 b 0.085 II 0.040c 0.36 2.5 1.50E-04 0.30 c 0.090 III 0.035d 0.36 2.5 2.30E-04 1.50 d 0.085 IV 0.035e 0.30 2.5 9.00E-04 1.75 e 0.080f 0.60 2.5 1.20E-03 2.75 f 0.080g 0.64 2.5 1.40E-03 3.50 g 0.080


870

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Ret

urn

flow

(m3/

s)

p arab olic curvep oints from the d is tr ib u ted T O PKAPI m od el

Ret

urn

flow

(m3 s-1

)

Fig. 5. The Beta-distribution function fitted to the distributed TOPKAPI simulated results (r=0.11,s=0.16)

Fig. 6. A limiting parabolic curve used for calculating the return flow from the soil water storage in the catchment

(32

2

1 aVV

aVV

aQss

sT

ss

sTreturn ++

=

with a1=441, a2=-330, a3=66.)

Table 3. Lumped TOPKAPI model parameters used in the Upper Reno river basin

Three non-linear reservoirs Beta-distribution function Parabolic curve function(used for calculating direct runoff) (used for calculating return flow)

Soil bs Surface bo Channel bc r s a1 a2 a37.98E-022 6.10E-010 7.35E-010 0.11 0.16 441. -330. 66.0


871

Fig. 7. Comparison of calculated runoff using the distributed and lumped TOPKAPI models for the Upper Reno catchment. Thick solid line:distributed model, thin solid line: lumped model. Major flood events.

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T im e

Run

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Run

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)

Fig. 8. Comparison of observed and modelled discharges for the Reno at Casalecchio. Thick solid line: observed; thin solid line: distributedmodel; dash line: lumped model. Major flood events.

Discussion of results

Figure 6 shows only marginal differences between thesimulated runoff (i.e. the total drainage into the channel)for the whole river basin and for the discharge atCasavecchio, generated by the distributed TOPKAPI modeland the lumped model, are practically identical with onlymarginal differences. This is despite the first one beingcomputed by integrating a cascade of 6325 non-linearreservoirs representing the soil, the overland flow and non-linear reservoirs representing the channel flow, whilst thesecond one is computed using the Beta-distribution functioncurve given by Eqn. (17) and a return-flow estimation curve

given by Eqn. (19), plus one soil reservoir simulating theinterflow (see Fig. 2), one surface reservoir simulating theoverland flow and one channel reservoir simulating thechannel flow, with an enormous reduction in computer time.

This case study demonstrates that the lumped version ofthe TOPKAPI model is already suitable for reproducing theoverall runoff that reaches the river network generated bythe distributed model. As far as the drainage networkcomponent is concerned, the lumped model produces a floodwave generally anticipated with respect to the distributedone (see Fig. 8), especially for the low flow period, whichis consistent with the non-linear behaviour of unsteady flowwith travel times reducing with the dimension of the event.


872

The TOPKAPI approach is attractive in that it is acomprehensive distributed/lumped approach in which thelumped model parameters are estimated directly from thedistributed model without calibration.

In this study, it was recognised that the phenomenon ofreturn flow identified in the distributed model must also beconsidered in the lumped model. Figures 9 and 10 show thecomparison of the simulation results obtained by the lumpedTOPKAPI model with and without consideration of returnflow. If the return flow was not considered in the lumpedmodel, the dynamics of the soil water storage and thecalculated discharges would be significantly different fromthose generated by the distributed model.

The Arno river basin case studyRIVER BASIN CHARACTERISTICS

The Arno river basin (Fig. 11a) drains an area of 8228 km2,rising in the mountain Falterona (1654 m a.s.l) situated inthe northern border of Casentino. The river flows in a south-west to north-east direction to the confluence with the Sieveriver (836 km2) where it flows east to west to the river mouth.Over its course, the Arno river is joined by major tributariesincluding the Greve river, Pesa stream, Elsa river and Erariver on the left side, and the Bisenzio river and Omroneriver on the right side. The watercourse of the river has atotal length of approximately 245 km. Local climatic

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Fig. 9. Comparison of observed and modelled discharges for Casalecchio. Thick solid line: observed; dash line: lumped model with returnflow; thin solid line: lumped model without consideration of return flow. Major flood events.

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873

FFiirreennzzee

SS.. GGiioovvaannnnii FFeerrrroovviiaa

FFoorrnnaacciinnaa

PPeessaa

SSuubbbbiiaannoo

NNaavvee ddii RRoossaannoo

Sieve

Casentino

Chiana

ValdarnoSuperiore

Elsa

Era

PaduleFucecchio

OmbronePistoiese

Bisenzio

GrevePesa

LLaatteerriinnaa

a) b)

c) d)

Fig. 11. Arno river basin: (a) river system, sub-basins and water level stations, (b) DEM map (legend: elevation above sea level in meter),(c) Soil type map, and (d) Land use map

tendencies produce the highest flooding risk in the periodfrom September to January when winds from the south-westdominate.

Topography

The highest areas are found in the montains of Falteronaand Pratomagno in the Casentino (Fig. 11b). The meanelevation of the whole river basin is 292 m a.s. l.. A DEM

data file with a grid scale of 200 m is available for the Arnoriver basin. In this study, the DEM is used with a grid sizeof 1km (see Fig. 11b).

Soils

Based on the SCS classification (Soil Conservation Service,1972), a soil property map (see Fig. 11c and Table 4) isavailable giving hydrological soil types. For the whole

Table 4. Legend for pedological data (soil hydrological types)

No. Hydrological DESCRIPTIONtype of soil(S.C.S/C.N.)

1 A With limited runoff potentiality. Contains deep sands with a very little lime and clay, and deep gravel of highpermeability.

2 B With moderately low runoff potentiality. Contains the major part of the sandy soil that is less deep than the onein the group A, but the group as a whole has a high infiltration capacity even when saturated.

3 C With moderately high runoff potentiality. Contains thin soils having a considerable quantity of clay and colloidalmaterial, while less than group D. The group has limited infiltration capacity when saturated.

4 D With middle high runoff potentiality. Contains the major part of the clay soil with a high capacity for expanding,but also the thin soils almost impermeable near the surface.


874

basin, the major soil type is C, which comprises 57.9 % ofthe total area. This implies that more than half of the soilspossess the hydrological characteristics of a thin soil layerwith moderately-high runoff potential, containing aconsiderable quantity of clay and colloidal material, withlimited infiltration capacity when saturated. Also, the soilsare thin and of lower permeability in the mountain areas,while the soils in low-lying areas are thick and of moderatelyhigh permeability.

Vegetation and land use

The primary land uses in this catchment (see Table 5) arethe classes of B (38.6%) and S (34.7%). In the mountainareas, most land use comprises shrub, undergrowth andovergrowth bush, while in the low-lying areas most land issuitable for arable farming.

di Rosano (Valdarno Superiore) and Firenze (Arno River)— are computed from hourly river levels by means of well-verified rating curves. The areal rainfall distribution wasestimated using the Thiessen Polygon method.

Calibration of the TOPKAPI model

The model calibration was performed at a 1-hour time-stepusing the hydrological dataset of 1 July to 31 December1992. The initial parameter values of the three basiccomponents in the TOPKAPI model, the soil water, surfacewater and channel water components, were estimated fromthe literature (e.g. using the USDA parameter tables for theGreen-Ampt infiltration model). The final parameter valueswere obtained by ‘trial and error’ on the basis of curvefitting.

Figures 12 to 16 show the comparison of the observeddischarges and those computed at Firenze, Nave di Rosano,Fornacina, Subbiano and Ponte della Ferrovia, respectively,using the distributed TOPKAPI model. The modelcalibration performance statistics and the parameter valuesare listed in Table 6 and Table 7, respectively.

Validation of the TOPKAPI model

Using the parameter values of Table 7, simulations werecarried out using the dataset for the period from 1 July to 31December 2000. Comparison of observed and simulatedhydrographs for the five stations is shown in Figs. 17 to 21.The model simulation performance statistics are shown inTable 8.

Discussion of results

Figures 12 to 21 demonstrate that the distributed TOPKAPImodel (based on a DEM with a large grid size of 1000 m)performed well in simulating the floods in the Arno riverbasin. It did not simulate well the low flows mainly due tothe absence of groundwater and lake/reservoir componentsin the present model. The oscillations at low flows shownin Figs. 12, 13, 17 and 18 for Firenze and Nave di Rosanoreflect the effect of a reservoir located at Laterina (see Fig.11).

Tables 6 and 8 show that the model performance in termsof the coefficient of determination (r2) in the validationperiod is lower than that in the calibration period, especiallyfor Fornacina and Subbiano. This is very probably due tothe poor quality of the measured water levels at the stationsof Fornacina and Subbiano. For event 2 (11 July 2000) themeasurements for Fornacina are significantly in error,because its peak is higher than the corresponding peaks at

Table 5. Legend for land use data

No. Class Description

1 U Urban areas with continuous tissue2 U1 Discontinuous urban areas3 CA Forest and arboreal vegetation4 B Vegetation of shrub, undergrowth and

overgrowth bush5 C Herbaceous vegetation, meadow-pasture6 CS Special cultivated areas, olive, vineyard7 S Suitable for sowing8 NV Areas without vegetation9 P Humid areas

APPLICATION OF THE DISTRIBUTED TOPKAPIMODEL

Data availablity

Floods have been experienced in 1992, 1993 and 2000. Itwas decided to choose 1992 as a flood year for modelcalibration, and use the dataset for 2000 for modelverification. Since most raingauges were concentrated inthe Upper Firenze in 1992, it was decided to calibrate theTOPKAPI model only in the catchment to Firenze with anarea of about 4325 km2.

Hourly measurements from 59 raingauges and 7temperature stations are available for 1992 and 2000, whilethe discharges of 5 stations — namely Fornacina (Sieve),Subbiano (Casentino), Ponte della Ferrovia (Chiana), Nave


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char

ge (m

3/s)

Dis

char

ge (m

3 s-1)

Fig. 12. Model calibration results compared with observed discharges at Firenze. Solid line: observed;Dash line: calculated. Major flood events.

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char

ge (

m3/

sec)

Dis

char

ge (m

3 s-1)

Fig. 13. Model calibration results compared with observed discharges at Nave di Rosano.

Fig. 14. Model calibration results compared with observed discharges at Fornacina.


876

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m3/

sec)

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char

ge (m

3 s-1)

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D a te

Dis

char

ge (m

3/s)

Dis

char

ge (m

3 s-1)

Fig. 15. Model calibration results compard with observed discharges at Subbiano.

Fig. 17. Model validation results compared with observed discharges at Firenze. Solid line: observed; Dash line: calculated.Major flood events.

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char

ge (

m3/

sec)

Dis

char

ge (m

3 s-1)

Fig. 16. Model calibration results compared with observed discharges at Ponte di Ferrovia.


877

Fig. 18. Model validation results compared with observed discharges at Nave di Rosano.

Fig. 19. Model validation results compared with observed discharges at Fornacina.

0

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£/s)

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char

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Dis

char

ge (m

3/s)

Dis

char

ge (m3 s-1

)

Fig. 20. Model validation results compared with observed discharges at Subbiano.


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D ate

Dis

char

ge (m

3/s)

Dis

char

ge (m

3 s-1)

Fig. 21. Model validation results compared with observed discharges at Ponte di Ferrovia.

Table 6. TOPKAPI model calibration statistics in the Upper Firenze catchment in 1992

Station Fornacina Subbiano Ponte delle Nave di FirenzePerformance Ferrovia Rosanostatistics

R2 0.875 0.845 0.855 0.915 0.916 r2 0.873 0.837 0.847 0.912 0.910 r 0.934 0.915 0.920 0.955 0.954

Table 7. Calibration parameters for the different soil classes, land uses and channel orders

Soil rs ϑϑ − α sk L Land use on Strahler cnclass (m s-1) (m) class (m-1/3 s-1) channel order (m-1/3 s-1)

A 0.500 2.5 6.54E-04 1.20 U 0.060 I 0.045B 0.535 2.5 6.06E-05 0.90 U1 0.100 II 0.040C 0.425 2.5 1.89E-05 0.60 CA 0.200 III 0.035D 0.417 2.5 1.67E-06 0.30 B 0.150 IV 0.030

C 0.100 V 0.025CS 0.090S 0.150NV 0.100P 0.010

Table 8. TOPKAPI model validation performance statistics for the Upper Firenze catchment in 2000

Station Fornacina Subbiano Ponte delle Nave di FirenzePerformance statistics Ferrovia Rosano

R2 0.739 0.626 0.820 0.891 0.878r2 0.732 0.608 0.811 0.889 0.858r 0.856 0.780 0.901 0.943 0.926


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Nave di Rosano and Firenze (see Figs. 19, 17 and 18), whichis generally impossible. Based on the measurements from 1July to 31 December 2000, the runoff coefficient forSubbiano is calculated to be 0.36 which is lower than itsgeneral value (greater than 0.50).

Conclusions and suggestionsA correct integration of the differential equations from thepoint to the finite dimension of a pixel, and from the pixelto larger scales, can actually generate relatively scaleindependent models, which preserve, as averages, thephysical meaning of the model parameters. Thisconsideration is reflected in the TOPKAPI approach. TheTOPKAPI model couples the kinematic approach with thetopography of the catchment and transfers the rainfall-runoffprocesses into three ‘structurally-similar’ non-linearreservoir equations describing different hydrological andhydraulic processes. The parameter values of the TOPKAPImodel are shown to be scale independent and obtainablefrom digital elevation maps (e.g. DTM, DEM), soil mapsand vegetation or land-use maps in terms of slope, soilpermeability, roughness and topology.

The TOPKAPI approach is a comprehensive distributed-lumped one. The distributed TOPKAPI is used to identifythe mechanism governing the dynamics of the saturated areacontributing to the surface runoff as a function of the totalwater storage. Theoretically it proves the lumped versionof the TOPKAPI model can be derived directly from thedistributed version and does not require additionalcalibration.

With the advantage of being a physically-based modelwith a simple and parsimonious parameterisation, theTOPKAPI model can have numerous applications rangingfrom flood forecasting, extreme flood analysis andpredicting hydrological response under the changedlandscape conditions caused by human activities. Finally,there exists an attractive possibility of deriving modelrepresentations from world 1 × 1 km2 cartography (such asfor instance GOTOPO30 produced by the USGS) to belumped at the 50 × 50 or 20 × 20 km2 meshes of theMesoscale or Limited Area Meteorological (LAM) Modelsthe better to reproduce the soil atmosphere exchanges.

The distributed version of the TOPKAPI model allowsfor its calibration on the basis of physical considerationsand, in particular, its extension to ungauged catchments.The lumped version of the TOPKAPI model allows for theextensive simulations needed when used in combinationwith a stochastic rainfall generator for deriving, throughcontinuous simulation, extreme discharges and flood wavevolumes.

The two applications of the TOPKAPI model to the UpperReno catchment and the Arno River basin demonstrate thatthe model performs well in simulating floods, although lowflows are not as well simulated. The model runs efficientlyin terms of calibrating and running time. For the Arno Riverbasin with 8315 cells, only 9 minutes are needed to run a 1-year period with input data sampled at 1 hour. The twoapplications also show that the model structure is simpleand reasonable in simulating the hydrological and hydraulicprocesses in a basin by using the non-linear reservoirapproximation.

Percolation to deeper soil layers has still to be introduced;this was ignored in the initial stage since it was not importantin the basins to which the model was originally applied.This objective may be pursued through the introduction ofa second soil layer with different characteristics from theupper layer, and involving water movement in a verticaldirection feeding into the aquifer. In addition, an approachfor incorporating lakes and reservoirs should be includedin further developments of the TOPKAPI model.

ReferencesAbbott, M.B., Bathurst, J.C., Cunge, J.A., O’Connell, P.E. and

Rasmussen, J., 1986a. An introduction to th EuropeanHydroloigcal system – Système Hydrologique Européen, SHE,1: History and philosophy of a physically based distributedmodelling system. J. Hydrol., 87, 45–59.

Abbott, M.B., Bathurst, J.C., Cunge, J.A., O’Connell, P.E. andRasmussen, J., 1986b. An introduction to th EuropeanHydroloigcal system – Système Hydrologique Européen, SHE,2: Structure of a physically based distributed modelling system.J. Hydrol., 87, 61–77.

Band, L.E., 1986. Topographic partition of watersheds with digitalelevation models. Water Resour. Res., 22, 15–24.

Barnes, H.H., 1967. Roughness characteristics of natural channels.United States Government Printing Office, Washington.

Benning, R.G., 1994. Towards a new lumped parameterisation atcatchment scale. Master Thesis, University of Wageningen, TheNetherlands. 10–45.

Beven, K.J., 1981. Kinematic subsurface stormflow. Water Resour.Res., 17, 1419–1424.

Beven, K.J., 1989. Changing ideas in hydrology – The case ofphysically based models. J. Hydrology, 105, 157–172.

Beven, K.J. and Kirkby, M.J., 1979. A physically based, variablecontributing area model of basin hydrology. Hydrol. Sci. Bull.,24, 1–3.

Beven, K.J., Kirkby, M.J., Schofield, N. and Tagg, A.F., 1984.Tesing a physically-based flood forecasting model(TOPMODEL) for three U.K. catchments. J. Hydrol., 69, 119–143.

Cash, J.R. and Karp, A. H., 1990. A variable order Runge-Kuttamethod for initial value problems with rapidly varying righthand sides. ACM Trans. Math. Software, 16, 201–222.

Chow, V.T., 1959. Open-Channel Hydraulics. International studentedition. McGraw-Hill.

Chow, V.T., Maidment, D.R., and Mays, L.W. 1988. AppliedHydrology. McGraw-Hill.


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de Marsily, G., 1986. Geostatistic and stochastic approach inhydrogeology. Chapter 11 in Quantitative hydrology:Groundwater Hydrology for Engineers. Academic Press, Inc.,California. 286_329.

Doorembos, J., Pruitt, W.O., Aboukhaled, A., Damagnez, J.,Dastane, N.G., van den Berg, C., Rijtema, P.E., Ashford, O.M.and Frere, M., 1984. Guidelines for predicting crop waterrequirements. FAO Irrig. Drainage Pap., 24.

Dunne, T., 1978. Field studies of hillslope flow process. In:Hillslope Hydrology M.J. Kirkby (Ed.). Wiley, New York. 227–293.

Franchini, M., Wendling, J., Obled, C. and Todini, E., 1996.Physical interpretation and sensitivity analysis of theTOPMODEL. J. Hydrol., 175, 293–338.

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Liu, Z., 2002. Toward A Comprehensive Distributed/LumpedRainfall-Runoff Model: Analysis of Available Physically-BasedModels and Proposal of a New TOPKAPI Model. Ph.D.dissertation, University of Bologna.

Liu, Z. and Todini, E., 2000. Application of TOPKAPI Model forExtreme Flood Frequency Analysis Study in Magra Catchment.Hydro-Geological Disasters Reduction: Recent DevelopmentsAnd Perspectives, Proc. Int. IDNDR Symposium held in Perugiain July.

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AppendixANALYTICAL SOLUTION TECHNIQUE FOR THENON-LINEAR RESERVOIR EQUATION

A general form of the non-linear reservoir equations in theTOPKAPI model can be written as:

cbyadtdy −= (A.1)

where y is a state variable (e.g. the average soil moisturecontent, soil water volume in the reservoir, or water depthover the slopes, or in the channel); a, b and c are constants(a may be equal zero) in each time-step.

According to the following three different cases:(a) a = 0; (b) 0≠a , 21 ≤≤ c ; and (c) 0≠a , 2>c , differenttechniques are used for solving the non-linear reservoirequation.

(a) a = 0. In this case, Eqn. (A.1) can be simplified to:

cbydtdy −= (A.2)

The solution to the above equation can be obtained directlyby:

[ ] cct ttcbyy −− −−+= 1

1

010 ))(1( (A.3)

where yt is the y at time t, y0 and t0 are the initial values of yand t at each time step.

(b) 0≠a , 21 ≤≤ c (e.g. 3/5=c for the overlandflow and channel flow equations of the TOPKAPI model).By taking the approximation of )( yyy c βα += , in which αand β can be estimated by using a least square method, Eqn.(A.1) is rewritten as:

)( ybyadtdy βα +−= (A.4)

))(( 2yy

bab

dtdy ++−−=

βα

ββ

(A.5)

Assuming βbA −= , βα=B and

Aa

baC =−=β

, integratingEqn. (A.5) gives


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∫∫ =

++t

t

y

ydtAdy

CByyt

002

1 (A.6)

If 1p and 2p are the two roots of the equation02 =++ CByy , then

02

42

1 ≥−+−= CBBp(A.7)

02

42

2 ≤−−−= CBBp(A.8)

When 10 py ≥ (i.e. for the flow recession period), theleft side of Eqn. (A.6) is rewritten as:

(A.9)

The analytical solution to Eqn. (A.6) is therefore obtainedby:

(A.10)

∫∫ −−

−−=

++tt y

y

y

ydy

pypyppdy

CByy 00

)11)(1(1

21212

( )1

20

10021212 )(1)(

−

−−

−−−−+=pypyttppApppyt

( )1

01

20021211 )(1)(

−

−−

−−−−−=yppy

ttppApppyt

1)1()1( −−−−= cc

ucacbdtdu

When 10 py < (i.e. for the flow rising period), similarlythe analytical solution to Eqn. (A.6) is obtained by:

(A.11)

(c) 0≠a , 2>c (e.g. c = 2.0 ~ 4.0 for the subsurface flowequation in TOPKAPI). By taking the substitution of

)1( −−= cyu , Eqn. (A.1) is rewritten as:

(A.12)

Since the term of 1−c

c in Eqn. (A.12) falls in the range1~2 now, it solution can be obtained analytically based onthe approximation described above.


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Towards a comprehensive physically-based rainfall-runoff model

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